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'''Rodan''' is a [[temperament]] of the [[gamelismic clan]] and one of the notable [[extension]]s to [[slendric]]. Like slendric, it is generated by an [[8/7]], three of which gives [[3/2]]. In addition it finds [[5/1|harmonic 5]] at +17 generator steps, and the second defining comma for it is [[245/243]].  
'''Rodan''' is a [[temperament]] of the [[gamelismic clan]] and one of the notable [[extension]]s to [[slendric]]. Like slendric, it is generated by an [[8/7]], three of which gives [[3/2]]. In addition it finds [[5/1|harmonic 5]] at +17 generator steps, and the second defining comma for it is [[245/243]].  


A reasonable tuning would be between [[41edo]] and [[46edo]]. [[87edo]] makes for a very recommendable option.  
Reasonable tunings would be between [[41edo]] and [[46edo]]. [[87edo]] makes for a very recommendable option.  


See [[Gamelismic clan #Rodan]] for more information.  
See [[Gamelismic clan #Rodan]] for more information.  
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{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | #
! rowspan="2" | Cents*
! rowspan="2" | Cents*
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|  
|  
|}
|}
<nowiki>*</nowiki> in 13-limit POTE tuning
<nowiki />* In 13-limit POTE tuning


== Notation ==
== Notation ==
A notation for rodan is listed in the notation guide for rank-2 pergens under [[pergen]] #8, (P8, P5/3). The generator is an upmajor 2nd. The [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is a trudminor 2nd, thus three ups equals a diatonic semitone, and three generators equals a perfect 5th. In rodan in particular, ^1 equals ~81/80 and ~64/63, and ^^1 equals ~33/32 and ~[[1053/1024]].  
A notation for rodan is listed in the notation guide for rank-2 pergens under [[pergen]]&nbsp;&#35;8, (P8,&nbsp;P5/3). The generator is an upmajor 2nd. The [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is a trudminor 2nd, thus three ups equals a diatonic semitone, and three generators equals a perfect 5th. In rodan in particular, ^1 equals ~81/80 and ~64/63, and ^^1 equals ~33/32 and ~[[1053/1024]].  


{| class="wikitable center-1 center-3"
{| class="wikitable center-1 center-3"
|+Rodan nomenclature<br>for selected intervals
|+ style="font-size: 105%;" | Rodan nomenclature<br />for selected intervals
|-
! Ratio
! Ratio
! Nominal
! Nominal
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| 3/2
| 3/2
| Perfect fifth
| Perfect fifth
| C-G
| C−G
|-
|-
| 5/4
| 5/4
| Downmajor third
| Downmajor third
| C-vE
| C−vE
|-
|-
| 7/4
| 7/4
| Downminor seventh
| Downminor seventh
| C-vBb
| C−vB♭
|-
|-
| 11/8
| 11/8
| Dup fourth
| Dup fourth
| C-^^F
| C−^^F
|-
|-
| 13/8
| 13/8
| Dupminor sixth
| Dupminor sixth
| C-^^Ab
| C−^^A♭
|}
|}
Rodan's notation has much in common with that for 41edo and 46edo, since both edos map a minor 2nd to three edosteps. It also resembles the notation for [[cassandra]]. All four notations notate the slendric tetrad (1-8/7-21/16-3/2) on C as C-^D-vF-G, and all four notations notate 5/4, 7/4, 11/8 and 13/8 as in the table above. But the notations diverge for other intervals, for example 11/10.
 
Rodan's notation has much in common with that for 41edo and 46edo, since both edos map a minor 2nd to three edosteps. It also resembles the notation for [[cassandra]]. All four notations notate the slendric tetrad ({{dash|1, 8/7, 21/16, 3/2|med}}) on C as {{nowrap|C, ^D, vF, G|med}}, and all four notations notate 5/4, 7/4, 11/8, and 13/8 as in the table above. But the notations diverge for other intervals, such as 11/10.


== Chords ==
== Chords ==
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{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>Generator
! Edo<br />Generator
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]
! Fifth<br>(¢)
! Fifth<br />(¢)
! Comments
! Comments
|-
|-

Revision as of 22:37, 1 January 2025

Rodan is a temperament of the gamelismic clan and one of the notable extensions to slendric. Like slendric, it is generated by an 8/7, three of which gives 3/2. In addition it finds harmonic 5 at +17 generator steps, and the second defining comma for it is 245/243.

Reasonable tunings would be between 41edo and 46edo. 87edo makes for a very recommendable option.

See Gamelismic clan #Rodan for more information.

Interval chain

Prime harmonics are in bold.

# Cents* Approximate Ratios
13-limit 17-limit Extension
0 0.000 1/1
1 234.482 8/7
2 468.964 21/16 17/13
3 703.446 3/2
4 937.929 12/7
5 1172.411 63/32, 160/81
6 206.893 9/8
7 441.375 9/7 22/17
8 675.857 40/27
9 910.339 22/13, 27/16, 33/28
10 1144.821 27/14, 35/18, 64/33 33/17
11 179.304 10/9
12 413.786 14/11, 33/26, 80/63
13 648.268 16/11
14 882.750 5/3
15 1117.232 21/11, 40/21
16 151.714 12/11
17 386.196 5/4
18 620.679 10/7
19 855.161 18/11, 64/39 28/17
20 1089.643 15/8 32/17
21 124.125 14/13, 15/14
22 358.607 16/13, 27/22 21/17
23 593.088 45/32 24/17
24 827.570 21/13
25 1062.052 24/13

* In 13-limit POTE tuning

Notation

A notation for rodan is listed in the notation guide for rank-2 pergens under pergen #8, (P8, P5/3). The generator is an upmajor 2nd. The enharmonic unison is a trudminor 2nd, thus three ups equals a diatonic semitone, and three generators equals a perfect 5th. In rodan in particular, ^1 equals ~81/80 and ~64/63, and ^^1 equals ~33/32 and ~1053/1024.

Rodan nomenclature
for selected intervals
Ratio Nominal Example
3/2 Perfect fifth C−G
5/4 Downmajor third C−vE
7/4 Downminor seventh C−vB♭
11/8 Dup fourth C−^^F
13/8 Dupminor sixth C−^^A♭

Rodan's notation has much in common with that for 41edo and 46edo, since both edos map a minor 2nd to three edosteps. It also resembles the notation for cassandra. All four notations notate the slendric tetrad (1 – 8/7 – 21/16 – 3/2) on C as C, ^D, vF, G, med, and all four notations notate 5/4, 7/4, 11/8, and 13/8 as in the table above. But the notations diverge for other intervals, such as 11/10.

Chords

Main article: Chords of rodan

Scales

Tuning spectrum

Gencom: [2 8/7; 154/153 196/195 245/243 273/272 364/363]

Gencom mapping: [⟨1 1 -1 3 6 8 8], ⟨0 3 17 -1 -13 -22 -20]]

Edo
Generator 		Eigenmonzo
(Unchanged-interval) 		Fifth
(¢) 		Comments
8/7 693.522
17/13 696.642
7/6 699.847
21\36 700.000 Lower bound of 7- and 9-odd-limit diamond monotone
9/7 700.750
4/3 701.955
24\41 702.439 Lower bound of 11- to 17-odd-limit diamond monotone
15/14 702.778
7/5 702.915 7- and 9-odd-limit minimax
11/9 703.041 11-odd-limit minimax
75\128 703.125
18/13 703.220 13- and 15-odd-limit minimax
16/15 703.240
12/11 703.244
15/11 703.359
13/12 703.371
15/13 703.410
51\87 703.448
5/4 703.467 5-odd-limit minimax
11/10 703.500
13/10 703.522
11/8 703.542
16/13 703.564
13/11 703.597
18/17 703.726 17-odd-limit minimax
17/15 703.748
78\133 703.759
6/5 703.791
20/17 703.894
24/17 703.956
17/16 704.257
10/9 704.292
27\46 704.348 Upper bound of 11- to 17-odd-limit diamond monotone
14/11 704.377
3\5 720.000 Upper bound of 7- and 9-odd-limit diamond monotone

Music

Gene Ward Smith
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